I decided to take a look at wikipedia. It reminded me of the integral test.

The sum of the first n terms > integral of 1/x dx from 1 to k+1 = ln(k+1)

Which implies the solution to ln(k+1)=10 is a lower bound.k = e^10 -1 = 22025.46***edit this is an upper bound***

Below that on the wikipedia entry is the rate of divergence:The sum of the first n terms = ln(k) + EMC + errorEMC is the Euler-Masceroni Constant which is about .5772156649and the error term is about 1/(2k)

Using the lower bound for the k in the error term gives k=e^(10-EMC-1/(2*22025.46) = 12366.87

I am pretty sure that the error term is small enough to imply 12366 is the solution to the problem. (12367 would exceed 10)